Structural equation models (SEMs) report findings in three different ways. Understanding the way statistical significance is reported requires understanding the terminology of the model itself. Within the graphical display of the model there are boxes and arrows. Boxes represent observed data and the arrows represent assumed causation. Within the model a variable that receives a one-way directional influence from some other variable in the system is termed "endogenous", or is dependent. A variable that does not receive a directional influence from any other variable in the system is termed as "exogenous", or is independent-in this case, percent minority, percent poor, percent disabled and percent aged 65 and over. When interpreting SEMs the values attached to one-way arrows (or directional effects) are regression coefficients, whereas two-way arrows (nondirectional relationships) are correlation coefficients; regression coefficients and correlations comprise the "parameters" of the model. The regression coefficients and correlations measure the strength of the relationship between the variables. A regression coefficient of .70 or higher indicates a very strong relationship; .50 to .69 indicates a substantial relationship; .30 to .49 indicates a moderate relationship; .10 to .29 indicates a low relationship; .01 to .09 indicates a negligible relationship; and a value of 0 indicates no relationship.
Besides regression coefficients and correlations, SEMs also test the overall fit of the model. The narrative analyses use three measures of model fit to determine the overall quality of fit of the model. Another way of thinking about model fit is to view this as the test of model significance, thus, when the values of significance are met for the tests all relationships within the model are significant, and it is then their relative strengths which decides if there is a relationship or not.
The first measure of model fit is the Goodness-of-Fit Index (GFI). The GFI measures the relative amount of variance and covariance in the Sample covariance matrix that is jointly explained by the Population covariance matrix. The GFI values range from 0 - 1, with values close to 1 being indicative of good fit.
A second type of Goodness-of-Fit index used in the analysis can be classified as incremental or comparative indexes of fit. As with the GFI, incremental indexes of fit are based on a comparison of the hypothesized model against some standard. However, whereas this standard represents no model at all for the GFI, for the incremental indices, it represents a baseline model (typically the independence or null model). Comparative Fit Index (CFI) is useful in that it takes sample size into account. The CFI values range from 0 to 1, but whereas .90 was considered a good fit for GFI, a revised cutoff of .95 has recently been advised for CFI.
The final set of fit statistics used in the analysis focuses on the Root Mean Square of Error Approximation (RMSEA). This fit statistic has only recently been recognized as one of the most informative criteria for use in covariance structure modeling. The RMSEA takes into account the error of approximation in the population and asks the question "How well would the model, with unknown but optimally chosen parameter values, fit the population covariance matrix if it were available?" This discrepancy, as measured by the RMSEA, is expressed per degree of freedom, thus making the index sensitive to the number of estimated parameters in the model (i.e. the complexity of the model); values less than .05 indicate good fit, values between .08 and .1 indicate mediocre fit, and those greater than .1 indicate poor fit. It is also possible to use confidence intervals to assess the precision of RMSEA estimates; AMOS (the statistical program that is used to run the SEMs) reports a 90% interval around the RMSEA value.
Besides testing for model fit, SEMs also provide a measure of multicollinearity. In some cases, the model fits the data well, even though none of the independent variables has a statistically significant impact on the dependent variables. How is this possible? When two independent variables are highly correlated, they both convey essentially the same information. In this case, neither may contribute significantly to the model after the other one is included. But together they contribute a lot. If you removed both variables from the model the fit would be much worse. So the overall model fits the data well, but neither independent variable makes a significant contribution when it is added to your model. When this happens, the independent variables are collinear and the results show multicollinearity. With SEMs, a correlation of .80 between variables is indicative of multicollinearity.
If your goal is simply to predict that the independent variables will influence your dependent variables, then multicollinearity is not a problem. The predictions will still be accurate. If your goal is to understand how the various independent variables impact the dependent variables, then multicollinearity is a big problem. The primary problem is that the individual strength values can be misleading (a strength value can be low, even though the variable is important). The best solution is to understand the cause of multicollinearity and remove it. Multicollinearity occurs because two (or more) variables are related - they measure essentially the same thing. If one of the variables doesn't seem logically essential to the model, removing it may reduce or eliminate multicollinearity. It is also possible to find a way to combine the variables. For example, if education, occupation and income were collinear independent variables, perhaps it would make scientific sense to remove education, occupation and income from the model, and use socio-economic status (calculated from education, occupation and income) instead. You can also reduce the impact of multicollinearity by increasing sample size.
